Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution
Abstract
:1. Introduction
2. Estimation of R in General Case
2.1. Maximum Likelihood Estimation
2.2. Asymptotic Distribution and Confidence Interval
2.3. Bootstrap Confidence Intervals for R
- Generate independent samples from PGHLD1, and from PGHLD2. The samples can be generated from (2) by sampling p from uniform distribution i.e., .
- Generate an independent bootstrap sample and taken with replacement from the given samples above in the first step. Based on the bootstrap sample compute the maximum likelihood estimates of say as well as the MLE of .
- Repeat step 2 to 3 B-times to obtain a set of bootstrap samples of R say , .
2.3.1. Percentile Bootstrap Confidence Interval :
2.3.2. Student’s t Bootstrap Confidence Interval ():
3. Estimation of R with Common Scale Parameter
3.1. Maximum Likelihood Estimation
3.2. Asymptotic Distribution and Confidence Intervals
4. Bayes Estimation of R
4.1. Bayes Estimation of R in General Case
- Step 1: Start with initial guess
- Step 2: Set
- Step 3: Use the Metropolis–Hastings algorithm to generate from and from
- Step 4: Use the Metropolis–Hastings algorithm to generate from and from
- Step 5: Compute from Equation (7)
- Step 6: Set
- Step 7: Repeat step 3 to 6, T times.
4.2. Bayes Estimation of R with Common Scale Parameter
- Step 1: Start with initial guess
- Step 2: Set
- Step 3: Use the Metropolis–Hastings algorithm to generate from and from
- Step 4: Use the Metropolis–Hastings algorithm to generate from
- Step 5: Compute from Equation (14)
- Step 6: Set
- Step 7: Repeat step 3 to 6, T times.
5. Simulation
6. Real Data Study
6.1. Real Data Study 1
6.2. Real Data Study 2
7. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
AEL | absolute error loss function |
ALCI | average length of confidence interval |
asymptotic confidence interval | |
BE | Bayes estimation |
percentile bootstrap confidence interval | |
student’s bootstrap confidence interval | |
CI | confidence interval |
CP | coverage probability |
GEL | general entropy loss function |
HPD | highest posterior density |
Fisher information matrix | |
KS | Kolmogorov-Smirnov |
L | log-likelihood |
LINEX | linear exponential loss function |
MAP | maximum a posteriori |
MLE | maximum likelihood estimation |
MSE | mean square error |
PHLD | Poisson half logistic distribution |
R | stress-strength parameter |
SEL | square error loss function |
Appendix A
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R | ||||||||||||||||||||
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Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
4 | ||||||||||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
R | ||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
Bias | MSE | Bias | MSE | Bias | MSE | Bias | MSE | |||||||||||||
ALCI | CP | ALCI | CP | ALCI | CP | ALCI | CP | |||||||||||||
L | |||||
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MLE | |||||
Bayes | − | ||||
KS | |||||
p-value |
R | ||||||
CI | ||||||
LCI |
L | |||||
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MLE | |||||
Bayes | − | ||||
KS | |||||
p-value |
R | ||||||
CI | ||||||
LCI |
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Muhammad, I.; Wang, X.; Li, C.; Yan, M.; Chang, M. Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution. Entropy 2020, 22, 1307. https://doi.org/10.3390/e22111307
Muhammad I, Wang X, Li C, Yan M, Chang M. Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution. Entropy. 2020; 22(11):1307. https://doi.org/10.3390/e22111307
Chicago/Turabian StyleMuhammad, Isyaku, Xingang Wang, Changyou Li, Mingming Yan, and Miaoxin Chang. 2020. "Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution" Entropy 22, no. 11: 1307. https://doi.org/10.3390/e22111307
APA StyleMuhammad, I., Wang, X., Li, C., Yan, M., & Chang, M. (2020). Estimation of the Reliability of a Stress–Strength System from Poisson Half Logistic Distribution. Entropy, 22(11), 1307. https://doi.org/10.3390/e22111307